Concluding Remarks

Recent sampling theory research on heteroskedastic models seems to be con­centrated on methods for estimation and hypothesis testing that do not requirespecification of a particular parametric form of heteroskedasticity. They are motivated by our inability to be certain about the most appropriate variance specification. However, methodology suggested along these lines is generally asymptotic and may not perform well in finite samples. What is likely to be important, and what seems to have been neglected, is whether the types of inferences we make in practice are very sensitive to the assumed form of hetero- skedasticity. If they are not, then efforts to develop alternative methods, that do not require an explicit variance function, may be misplaced.

Bayesian estimation has several advantages. Results are presented in terms of intuitively meaningful posterior pdfs. Marginal posterior pdfs reflect all the para­meter uncertainty in a model and do not condition on point estimates of nuisance parameters. Predictive pdfs for future values can also be constructed without conditioning on point estimates (Boscardin and Gelman, 1996). The advent of MCMC techniques means that many more practical applications of Bayesian inference to heteroskedastic models are now possible.

Note

* The author acknowledges valuable comments on an earlier version by three anony­mous reviewers.

References

Amemiya, T. (1973). Regression analysis when the variance of the dependent variable is proportional to the square of its expectation. Journal of the American Statistical Association 68, 928-34.

Griffiths, W. E., and J. R. Anderson (1982). Using time-series and cross-section data to estimate a production function with positive and negative marginal risks. Journal of the American Statistical Association 77, 529-36.